Abstract
Diffusively coupled chemical oscillators can exhibit a wide variety of complex spatial patterns. In this paper, we show that a ring of relaxation oscillators diffusively coupled through the inhibitory species leads to remarkable spatiotemporal patterns in the regime where there is a large separation of time scales between the activator and the inhibitor dynamics. The origin of these complex patterns can be traced back to a preponderance of antiphase synchronized states in the space of attractors. We provide an analytical explanation for the existence and stability of the antiphase synchronized states by examining the limit of extreme time scale separation. Numerical results on rings with small numbers of oscillators show that an explosion of patterns occurs for a ring with five oscillators.
A classic paradigm of morphogenesis is Turing's model of the embryo in which “cells” undergoing chemical reactions are coupled diffusively. A recent experimental realization of this theoretical model exhibits complex spatiotemporal patterns. Our work is motivated by these experiments and the broader question of the synchronization patterns of coupled, strongly nonlinear oscillators in a geometrically frustrated ring in which the preferred phase relationship cannot be satisfied for all nearest neighbors. We model the individual cells by the Brusselator, a well-studied model of autocatalytic reactions. However, we study it in the limit of a large time scale separation between the inhibitor and the activator dynamics. Coupling through the fast, inhibitory species leads to complex spatiotemporal patterns because of a preference for neighboring oscillators to be in antiphase relationship. An analytic calculation in the regime of extreme time scale separation demonstrates the origin of the antiphase synchronization.